A collection $\{f_n\}$ of real valued functions is said to be HK-equi-integrable on $I=[a,b]$, if there exists a gauge $\delta$ on $I$ such that for every $\epsilon>0$, there exists a $\delta$-fine tagged partition $\mathcal{P}$ of $I$ such that $\|S(f,\mathcal{P})-\int_I f\|_X<\epsilon$ for every $n$. That is, the gauge $\delta$ is independent on $n$.
Where gauge $\delta$ is any positive function $\delta:[a,b]\rightarrow\mathbb{R}^+$.
And $S(f,\mathcal{P})=\sum_i f(\xi_i)(x_i-x_{i-1})$.

Now my question is that does equi-integrability of $\{f_n\}$ and point wise convergence of $\{f_n\}$ to $f$ implies uniform convergence of the sequence $\{f_n\}$?

This question is a duplicate (slightly expanded) of mathoverflow.net/questions/72447 . @Pietro: The whole question is about Henstock–Kurzweil integral. What would be the point of asking $f_n$ to be Riemann integrable?
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Emil JeřábekAug 9 '11 at 12:06

no special point, but to have a clear question.
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Pietro MajerAug 9 '11 at 13:23

Oh I didn't notice all these clones of the question.
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Pietro MajerAug 9 '11 at 13:26

1

In the closed version of the question, Emil Jeřábek says: ... I think that the intended definition of HK-equi-integrability should read: for every $\epsilon>0$ there exists a gauge $\delta$ such that for every $n$ and every $\delta$-fine partition $\mathcal P$, $|S(f_n,\mathcal P)−\int f_n|\le \epsilon$.
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Gerald EdgarAug 9 '11 at 14:24